ISSN 1311-8080

IJPAM: Volume 64, No. 1 (2010)

Abstract.The central object of synthetic differential geometry is microlinear spaces.
In our previous paper (Microlinearity in Frölicher spaces - beyond the
regnant philosophy of manifolds, International Journal of Pure and Applied
Mathematics, 60 (2010), 15-24) we have emancipated microlinearity from within
well-adapted models to Frölicher spaces. Therein we have shown that
Frölicher spaces which are microlinear as well as Weil exponentiable form
a Cartesian closed category. To make sure that such Frölicher spaces are
the central object of infinite-dimensional differential geometry, we develop
the theory of vector fields on them in this paper. Our principal result is
that all vector fields on such a Frölicher space form a Lie algebra.